Sketching the graph of a function f (x)=2x3 − 3x2 − 12x +2 Calculus and Differential Equations I MATH 250 A Do you know how to sketch the graph of this function without using a calculator? 1 Yes Maxima, minima, inflection points, and differentiability 2 No If one adds a constant to the right hand side of the above equation, what does this do to the graph of f ? 1 Translate the graph horizontally 2 Translate the graph vertically Whatdoyouneedtodototranslate the graph horizontally? Maxima, minima, inflection points, and differentiability Calculus and Differential Equations I Maxima, minima, inflection points, and differentiability Calculus and Differential Equations I Continuity and differentiability Local and global extrema Can a function f be continuous but not differentiable at a point? A function f has a local maximum at a point x0 if 1 Yes f (x) ≤ f (x0)forx near x0. 2 No Similarly, a function f has a local minimum at a point x0 if Can a function f be differentiable but not continuous at a f (x) ≥ f (x0)forx near x0. point? 1 Yes An extremum (minimum or maximum) of a function can be 2 No local or global. If f is a smooth function, is it possible to have f positive and Does a function always have a global maximum? f locally decreasing? 1 Yes 1 Yes 2 No 2 No If f is a smooth function of x and f (x) > 0, then Is it possible for a function to have more than one global minimum? 1 f is concave up 1 2 f is concave down Yes 2 3 One cannot tell No Maxima, minima, inflection points, and differentiability Calculus and Differential Equations I Maxima, minima, inflection points, and differentiability Calculus and Differential Equations I Extrema and first derivative Extrema and first derivative (continued) If f is differentiable and f (x0) = 0, does it mean that f has a If a function f has a local minimum at x =0,doesit local maximum or minimum at x = x0? f necessarily mean that (0) = 0? 1 Yes 1 Yes 2 No 2 No First-derivative test: if f has a critical point at x = x0,is A function f has a critical point at x = x0 if either f (x0)=0 continuous at x = x0, and is differentiable on the right and on or f (x0) is undefined. the left of x0,then If f goes from negative to positive as x is increased past x0, f x x x Fact: If has a local maximum or minimum at = 0,if 0 is then f has a local minimum at x = x0. not an end-point of the domain of definition of f ,andiff is If f goes from positive to negative as x is increased past x0, differentiable at x = x0,thenf (x0)=0. then f has a local maximum at x = x0. We now consider the converse of this statement. What should In particular, if f is twice differentiable on some interval, if f we require of f for f to have a local maximum or a local has a constant sign near x0,andifx0 is not an end-point of minimum at x = x0? the interval, then f has a local maximum or minimum at x = x0 if and only if f (x0)=0. Maxima, minima, inflection points, and differentiability Calculus and Differential Equations I Maxima, minima, inflection points, and differentiability Calculus and Differential Equations I Extrema and second derivative, inflection points Inflection points and second derivative Second-derivative test: If f is twice-differentiable, if f is If f is twice-differentiable and x0 is not an end-point of the continuous, and if f (x0) = 0, then domain of definition of f , is the following statement true of If f (x0) > 0, then f has a local minimum at x0. false? If f (x0) < 0, then f has a local maximum at x0. Statement: If f (x0)=0,thenf has an inflection point at If f (x0) = 0, then one cannot conclude the existence of a x = x0. local extremum at x = x0. 1 True 2 False An inflection point of a function f is a point x0 such that the f x x graph of changes concavity at = 0. If f is twice-differentiable and if f changes sign when x goes x f x x In other words, the graph of the function f crosses its tangent through 0,then has an inflection point at = 0. at an inflection point. Reading assignment: Section 4.1 of the Calculus book. How are inflection points and second derivatives related? Homework: Review of differentiation in WebAssign. Maxima, minima, inflection points, and differentiability Calculus and Differential Equations I Maxima, minima, inflection points, and differentiability Calculus and Differential Equations I.